Solar Photovoltaic Array Perimeter & Inverter Enclosure (Civil & Construction Engineering)

In utility-scale solar engineering projects across the Mojave Desert in Southern Nevada, civil construction engineers design high-voltage photovoltaic (PV) array fields with secured perimeter fencing. Heavy security perimeter fencing with embedded grounding loops and fiber-optic intrusion detection costs significantly more per linear meter than internal partition chain-link fencing used to isolate central inverter substations.

A civil engineering team is designing a rectangular solar collection zone covering a total footprint area of $120,000\text{ m}^2$. To satisfy electrical safety codes, the field must be split into three identical adjacent rectangular inverter blocks using two internal dividing fence lines parallel to the shorter side $x$:

       ================ Outer Security Fence (y) ================
      |                       |                       |          |
      |   Inverter Block 1    |   Inverter Block 2    | Inverter | x
      |                       |                       | Block 3  |
      |                       |                       |          |
       ================ Outer Security Fence (y) ================

The outer perimeter fencing (the two horizontal sides of length $y$ and the two outer vertical sides of length $x$) costs $\$150$ per linear meter due to reinforced concrete trench footings, grounding copper wire, and security cameras. The two interior dividing fences (each of length $x$) cost only $\$50$ per linear meter because they do not require anti-intrusion trenching.

1. Formulate the total fencing cost function $C(x, y)$ and state the area constraint governing the solar field footprint.
2. Apply variable reduction using the Applied Optimization Modeling Framework to express total construction cost as a single-variable function $C(x)$.
3. Establish the Context-Driven Domain for the dividing width $x$.
4. Determine the exact dimensions ($x$ and $y$) that minimize total fencing construction cost, and compute the minimum total expenditure.
5. Apply the Second Derivative Test to mathematically prove that this critical point achieves an absolute minimum cost.